Simplify; express your answer in exponential form. Assume $r\neq 0, p\neq 0$. $\dfrac{{(r^{-4}p^{-4})^{-3}}}{{(r^{-3}p^{-1})^{3}}}$
Explanation: To start, try simplifying the numerator and the denominator independently. In the numerator, we can use the distributive property of exponents. ${(r^{-4}p^{-4})^{-3} = (r^{-4})^{-3}(p^{-4})^{-3}}$ On the left, we have ${r^{-4}}$ to the exponent ${-3}$ . Now ${-4 \times -3 = 12}$ , so ${(r^{-4})^{-3} = r^{12}}$ Apply the ideas above to simplify the equation. $\dfrac{{(r^{-4}p^{-4})^{-3}}}{{(r^{-3}p^{-1})^{3}}} = \dfrac{{r^{12}p^{12}}}{{r^{-9}p^{-3}}}$ Break up the equation by variable and simplify. $\dfrac{{r^{12}p^{12}}}{{r^{-9}p^{-3}}} = \dfrac{{r^{12}}}{{r^{-9}}} \cdot \dfrac{{p^{12}}}{{p^{-3}}} = r^{{12} - {(-9)}} \cdot p^{{12} - {(-3)}} = r^{21}p^{15}$